1.7 Midpoint and Distance in the Coordinate Plane

SOL: G3aObjectives: TSW …• To find the midpoint of a segment.• To find the distance between two points in the

coordinate plane.

Midpoint

the point that divides the segment into two congruent () segments.

XA B

Midpoint

If X is the midpoint of AB, then AX XB and AX = XB

How do we find the Midpoint

If we have a Number Line, then we use the two endpoints, add them together and divided by two.

MP Q

a b

Midpoint = a + b 2

Example 1: Finding the Midpoint

The coordinates on a number line of J and K are–12 and 16, respectively. Find the coordinate ofthe midpoint of JK.

J K

-12 16

Midpoint = -12 + 16 2

42

= = 2

Example 2: Finding the Midpoint

MN has the endpoints at -9 and 4. What is the coordinate of its midpoint?

M N

-9 4

Midpoint = -9 + 4 2

-5 2

= = -2 ½

Midpoint Formula: If we are working in the Coordinate Plane

When using the midpoint formula you can use the points in any order, remember addition is commutative.

8 + -14 2

Example 3:

Find the coordinates of the midpoint of GH for

G(8, -6) and H(-14, 12).

Midpoint = -6 2= ( , )-6 + 12

2( , )6

2 = (-3, 3)

( , )( , )

Example 4:

Find the coordinates of the midpoint of RS for

R(5, -10) and S(3, 6).

Midpoint = 5 + 3 2

82=

-10 + 6 2

-4 2 = (4, -2)

( , )

Example 5: Find the missing endpoint

Find the coordinates of D if E(-6, 4) is the midpoint of DF and F has coordinates (5, -3).

5 + x 2

-3 + y 2

(-6, 4) =

5 + x 2

-3 + y 2

-6 = 4 =

-12 = 5 + x-5 -5

-17 = x

8 = -3 + y+3 +3

11 = y(-17, 11) =

Example 6:

What is the measure of PR if Q is the midpoint of PR?

PQ = QR = 6 – 3x

(6 – 3x) + (6 – 3x) = 14x + 212 – 6x = 14x + 2

+ 6x + 6x12 = 20x + 2-2 - 2

10 = 20x

10 = 20x20 20

½ = x

PR = 14x + 2= 14(½) + 2= 9

Distance Formula – Coordinate Plane to find Distance

The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by

212

212 )()( yyxxd

Based on the Pythagorean Theorem, which we will learn about later.

Example 7: What is the distance?

Find the distance between E(-4, 1) and F(3, -1)

212

212 )()( yyxxd

EF = (-4 – 3)2 + (1 +1)2

EF = 49 + 4

EF = 53

EF ≈ 7.28

Example 8: What is the distance?

Find the distance between S(-2, 14) and R(4, 3)

212

212 )()( yyxxd

SR = (-2 – 4)2 + (14 - 3)2

EF = 36 + 144

EF = 180

EF ≈ 13.416